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What is a Pythagorean triple? What do Pythagorean triples have to do with
Fermat's Last Theorem?

A Pythagorean triple is a set of three positive whole numbers a,
b, and c that are the lengths of the sides of a right triangle. This
means that a, b, and c satisfy the equation from the
Pythagorean Theorem, namely

a2 + b2 = c2.

The smallest example is a = 3,b = 4, and c = 5.
You can check that

32 + 42 = 9 + 16 = 25 = 52.

Sometimes we use the notation (a,b,c) to denote such a triple.

Notice that the greatest common divisor of the three numbers 3, 4, and 5is 1. Pythagorean triples with this property are called
primitive. From primitive Pythagorean triples, you can get other,
imprimitive ones, by multiplying each of a, b, and c by any
positive whole number d > 1. This is because

a2 + b2 = c2 if and only if
(da)2 + (db)2 = (dc)2.

Thus (a,b,c) is a Pythagorean triple if and only if (da,db,dc) is. For
example, (6,8,10) and (9,12,15) are imprimitive Pythagorean triples.

Formulas for Primitive Pythagorean Triples and Their Derivation

Suppose that we start with a primitive Pythagorean triple (a,b,c). If
any two of a,b,c shared a common divisor d, then, using the equation

a2 + b2 = c2,

we could see that the d2 would have to divide the square of
the remaining one, so that d would have to divide all three. This would
contradict the assumption that our triple was primitive. Thus no two of
a, b, and c have a common divisor greater than 1.

Now notice that not all of a, b, and c can be odd, because if
a = 2x + 1,b = 2y + 1, and c = 2z + 1,
then the equation would be

This implies that 4 is a divisor of 2, which is also false. Thus
either a or b must be even, and the other two odd. Let's say it is
b that is even, with a and c odd. (If not, switch the meanings of a
and b in what follows below.)

Now rewrite the equation in the form

b2 = c2 - a2,
(b/2)2 = ([c - a]/2)([c + a]/2).

Notice that since b is even and a and c are odd, b/2, (c - a)/2, and
(c + a)/2 are whole numbers, and all are positive. Now we claim that the
greatest common divisor of (c - a)/2 and (c + a)/2is 1. If d is any common divisor of these, then d would divide their
sum, c, and their difference, a. We know, however, that the greatest common divisor
of a and cis 1, so d must divide 1, so d = 1.

This gives us the product of two whole numbers, (c - a)/2 and (c + a)/2,
whose greatest common divisor is 1, and whose product is a square. The
only way that can happen is if each of them is a square itself. This
means that there are positive whole numbers r and s such that

r2 = (c + a)/2,
s2 = (c - a)/2,
rs = b/2.

Furthermore, r > s, because r2 = s2 + a >
s2, and s > 0, because c > a.
In addition, since r2 and s2 have greatest common divisor 1, likewise
r and s have greatest common divisor 1. Lastly, r and s can't both be
odd, or else r2 - s2 = a would be even, which it
isn't, and they can't both be even since 2 can't be a common divisor.
This means that one of r and s is odd and one is even; hence r - s is odd.

Now, solving the above three equations for a, b, and c, we find that, for
primitive Pythagorean triples,

a = r2 - s2,

b = 2rs,

c = r2 + s2,

r > s > 0 are whole numbers,

r - s is odd, and

the greatest common divisor of r and s is 1.

To see that the a, b, and c defined by these formulas do form a
Pythagorean triple, just check the equation:

a2 + b2

=

(r2 -
s2)2 + (2rs)2,

=

r4 - 2r2s2 +
s4 + 4r2s2,

=

r4 + 2r2s2 +
s4,

=

(r2 +
s2)2,

=

c2.

The above formulas for a, b, and c are the most general formulas for
primitive Pythagorean triples. To every pair of whole numbers r and s
satisfying those conditions, there corresponds a primitive Pythagorean
triple, and to every Pythagorean triple, there corresponds a pair of
whole numbers r and s satisfying those conditions. To find
r and s given a, b, and c, use

r = sqrt([c + a]/2),
s = sqrt([c - a]/2).

Table of Small Primitive Pythagorean Triples

Here is a table of the first few primitive Pythagorean triples (Michael Somos provides
a larger Pythagorean Triple
Table on the Web):

r

s

a

b

c

2

1

3

4

5

3

2

5

12

13

4

1

15

8

17

4

3

7

24

25

5

2

21

20

29

5

4

9

40

41

6

1

35

12

37

6

5

11

60

61

7

2

45

28

53

Formulas for All Pythagorean Triples

To include all Pythagorean triples, both primitive and imprimitive, we
let d > 0 be a whole number, and set

a = (r2 - s2)d,

b = 2rsd,

c = (r2 + s2)d,

r > s > 0 and d > 0 are whole numbers,

r - s is odd, and

the greatest common divisor of r and s is 1.

These formulas represent every Pythagorean triple. Given a Pythagorean
Triple, we can recover d, r, and s using

d = GCD(a,b,c),
r = sqrt([c + a]/[2d]),
s = sqrt([c - a]/[2d]).

There is a one-to-one correspondence between Pythagorean triples and
sets of values of the three parameters r, s, and d satisfying the
conditions given above.

Perimeter, Area, Inradius, and Shortest Side

The perimeter P and area K of a Pythagorean triple triangle are given by

P = a + b + c = 2r(r + s)d,
K = ab/2 = rs(r2 - s2)d2.

The radius of the inscribed circle, or inradius, is always a whole
number, and is given by the formula s(r-s)d.

The equation an + bn = cn, n > 2,
a, b, and c positive whole numbers, has no solutions. This statement
is called "Fermat's Last Theorem."
In the 17th century Pierre de Fermat conjectured that this equation has no solutions.
This famous problem withstood the attacks of the mathematical world for more than
three centuries. It was finally proved in the 1990's by Andrew Wiles, with help
from Richard Taylor, using extremely advanced methods.